The existence of multi-peak positive solutions for nonlinear Kirchhoff equations

Abstract

In this work, we study the following Kirchhoff equation $$\{\begin{array}{l}-\left({\epsilon }^{2}a+\epsilon b{\int }_{{\mathbb{R}}^3}|\nabla u{|}^2\right)\Delta u+u=Q(x)u^{q-1},u>0,x\in {\mathbb{R}}^3,\\ u\to 0,\text{as}|x|\to +\infty ,\end{array}$$ where $a,b>0$ are constants, $2<q<6$, and $\epsilon >0$ is a parameter. Under some suitable assumptions on the function $Q(x)$, we obtain that the equation above has positive multi-peak solutions concentrating at a critical point of $Q(x)$ for $\epsilon >0$ sufficiently small, by using the Lyapunov-Schmidt reduction method. We extend the result in (Discrete Contin. Dynam. Systems 6(2000), 39--50) to the nonlinear Kirchhoff equation.